Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #48383 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,15,18)(11,19,21,16)(12,22,23,13)(14,17,20,24)(25,26,31,34)(27,32,35,30)(28,29,33,39)(36,37,40,38);(3,8)(6,7)(9,11,15,21)(10,16,18,19)(12,17,23,24)(13,20,22,14)(25,27,31,35)(26,32,34,30)(28,36,33,40)(29,37,39,38);(1,3,5,8)(2,6,4,7)(12,23)(13,22)(14,20)(17,24)(27,35)(28,33)(29,39)(30,32);(3,8)(6,7)(9,12,15,23)(10,13,18,22)(11,17,21,24)(14,16,20,19)(25,28)(26,29)(27,36)(30,38)(31,33)(32,37)(34,39)(35,40);(1,3)(2,7)(4,6)(5,8)(9,13)(10,12)(11,20)(14,21)(15,22)(16,17)(18,23)(19,24)(25,29)(26,33)(27,37)(28,34)(30,36)(31,39)(32,40)(35,38);(1,4,5,2)(3,6,8,7)(9,14,15,20)(10,17,18,24)(11,13,21,22)(12,16,23,19)(25,30)(26,27)(28,38)(29,36)(31,32)(33,37)(34,35)(39,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8889351);